In addition to transverse waves on a string, there are always longitudinal waves present as well. In fact, longitudinal waves hold all of the potential energy associated with the transverse waves, and they carry the forward momentum in the direction of propagation associated with transverse traveling waves [122,391]. Longitudinal waves in a string typically travel an order of magnitude faster than transverse waves on the same string and are only weakly affected by changes in string tension.
Longitudinal waves are often neglected, e.g., in violin acoustics, because they couple inefficiently to the body through the bridge, and because they are ``out of tune'' anyway. However, there exist stringed instruments, such as the Finnish Kantele , in which longitudinal waves are too important to neglect. In the piano, longitudinal waves are quite audible; to bring this out in a striking way, sound example 5 provided in [18, Conklin chapter] plays Yankee Doodle on the longitudinal modes of three piano strings all tuned to the same (transversal) pitch. The nonlinear nature of the coupling from transverse to longitudinal waves has been demonstrated in . Longitudinal waves have been included in some piano synthesis models [30,28,24,23].